Problem 8 You open a 5-year CD for $1,000 ... [FREE SOLUTION]

Chapter 17: Problem 8

You open a 5-year CD for $1,000 that pays 2% interest, compounded annually. What is the value of that CD at the end of the five years?

Short Answer

Expert verified

The value of the 5-year CD at the end of five years will be approximately $1,104.08.

Step by step solution

Identify the given information

In this problem, we are given the following information: - Initial deposit, P = $1,000 - Interest rate, r = 2% per year (0.02 as a decimal) - Time, t = 5 years - Compounding frequency, n = 1 (compounded annually)

Use the formula for compound interest

To find the value of the CD at the end of 5 years, we will use the formula for compound interest, which is: A = P(1 + r/n)^(nt) where: A = Value of the CD at the end of 5 years P = Initial deposit ($1,000) r = Interest rate (0.02) n = Compounding frequency (1, compounded annually) t = Time (5 years)

Plug the values into the compound interest formula

Now, let's plug the given values into the compound interest formula: A = $1,000(1 + 0.02/1)^(1 * 5)

Simplify the expression and solve for A

Let's simplify the expression and solve for A: A = $1,000(1 + 0.02)^5 A = $1,000(1.02)^5 Calculating the value of (1.02)^5, we get: A = $1,000 * 1.10408

Calculate the final value and present as a rounded value

Now we can calculate the final value of A and round it to a suitable number of decimal places (two decimal places, since we're dealing with dollars and cents): A = $1,104.08 The value of the 5-year CD at the end of five years will be approximately $1,104.08.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Certificate of Deposit (CD)

A Certificate of Deposit, commonly known as a CD, is a type of savings account offered by banks and credit unions. It provides a safe place to store your money and earn interest over a specified period of time.
Here's what you need to know about CDs:

Fixed Term: CDs have a set time duration, ranging from a few months to several years, such as the 5-year CD in our example.
Interest Rate: CDs typically offer a fixed interest rate, meaning the rate does not change over the duration of the term. This helps with planning the growth of your savings.
Early Withdrawal Penalty: Withdrawals before the term ends are usually penalized, so it's best to ensure you won't need the money during the term.
Safety: CDs are considered low-risk investments. They often come with insurance, such as FDIC guarantee for bank CDs, making them a safe option for your savings.

In summary, a CD is great for those who want a low-risk investment with a known return over a specific period. Its predictability makes financial planning easier!

Interest Rate

The interest rate is one of the most critical aspects when investing in a CD. It represents the percentage at which your investment grows year over year. In our example, the CD has an interest rate of 2% per annum.
There are a few key points about interest rates:

Fixed vs. Variable: A CD usually has a fixed interest rate, which means it remains the same throughout the entire term.
Affects Returns: The higher the interest rate, the more your investment grows over time.
Annual Compounding: In many CDs, interest is compounded annually, meaning the interest earned each year is added to the principal, and the next year's interest is calculated on the new total.
Market Influence: Interest rates are influenced by broader economic factors, including actions by central banks, inflation, and market demand.

A 2% interest rate compounded annually might not seem high, but it ensures steady growth without taking risks. Understanding how this rate affects your returns is crucial for financial planning.

Time Value of Money

The time value of money (TVM) is a core principle in finance. It suggests that a sum of money now is worth more than the same sum in the future. This concept is fundamental when deciding on investments like a CD.
It's based on a few key ideas:

Potential Earning: Money today can be invested to earn interest or returns, growing in value over time.
Inflation: Over time, the value of money decreases due to rising prices, which reduces its purchasing power.
Opportunity Cost: Holding onto money now might mean missing out on potential returns elsewhere, making its evaluation even more critical.
Discrepancy: A higher interest rate or longer investment period increases the future value of an initial sum, as seen in the compound interest formula used in our example.

Thus, TVM helps in comparing investment opportunities. It aids individuals in understanding that delayed access to money has a cost, emphasizing the importance of effective interest rates and time horizons in investment decisions.

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